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\begin{document}

The general variational method for estimating the ground state energy of a quantum mechanical system, or equivalently the equilibrium energy expectation value $\angle{E}$ for a statistical mechanical system, proceeds as follows:

\begin{enumerate}
\item Formulate an ansatz for the functional form of $\angle{E}$
\item Paramterise $\angle{E}$ by a set of parameters $\{\lambda\}$
\item Minimise $\angle{E}$ w.r.t. $\{\lambda\}$ by demanding that $$\nalba_\lambda \angle{E(\{\lambda\})} |_\{\lambda_0\} = 0$$
\item Compute $\angle{E(\{\lmbda_0\}) \geq \angle{E}$ i.e. the upper bound on $\angle{E}$
\end{enumerate}

The accuracy of this estimate can typically be improved by separating the hamiltonian into an exactly solvable part $\ham_0$, and a perturbation $\ham'$ $$\ham = \ham_0 + \ham'$$ then applying the variational method to $\ham'$. Often the full hamiltonian cannot be separated immediately. The prototypical example of this in statistical mechanics is Ising ferromagnet in 2D, where

$$\ham = - k \sum_\angle{\vec r,\vec r'} \sigma(\vec r) \sigma(\vec r'), k > 0$$

in these case we can re-write the hamiltonian according to

$$\ham = \ham_0 + (\ham - \ham_0)$$

where $\ham_0$ is exactly solvable, and we regard $\ham' := \ham - \ham_0$ as a pertubation; the so-called ``mean-field'' hamiltonian, to which we will apply the variational method.

For the 2D Ising ferromagnet we make the choice

$$\ham_0 = h \sum_\{\vec r\} \sigma(\vec r), h > 0$$

The resulting partition function then factorises according to

$$Z_N = \sum e^{-\beta ( \ham_0 + (\ham - \ham_0) )} = [ \sum e^{-\beta \ham_0} ] [ \sum e^{-\beta (\ham - \ham_0)} ] = Z_N^(0) \sum e^{- \beta (\ham - \ham_0)}$$

Where $Z_N^(0)$ is the partition function of a collection of $N$ non-interacting spins, i.e.

$$Z_N^(0) = \left[ \sum_{\sigma = \pm 1} e^{-\beta h \sigma} \right]^N = [ 2 \cosh \beta h ]^N$$

Now we may calculate the 


The exact action is

$$-\beta \ham = - K \sum_\angle{i,j} \sigma_i \sigma_j$$

we add a fictitious ``mean field'' term, such that

$$\diff{}{H} Z_N = \angle{\sigma} =: m$$

giving the action

$$-\beta \ham = -K \sum_\angle{i,j} \sigma_i \sigma_j + H \sum_i \sigma_i$$



\end{document}
